共查询到20条相似文献,搜索用时 578 毫秒
1.
V. S. Panferov 《Mathematical Notes》1973,14(5):936-942
Let ∥·∥ be a norm in R2 and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R2 rational if (x1 ? y1)/(x2 ? y2) or (x2 ? y2)/(x1 ? y1) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T2 of positive planar measure implies ∥T v ∥L4 (T2) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T v } and a set E ? T2, ¦E¦ > 0, such that T v (x) → 0(v → ∞) on E; however, ¦cn¦ ? 0 for ∥n∥ → ∞. 相似文献
2.
Let X be a Banach space and T:X → X a continuous map, which is expanding (i.e., ∥Tu ? Tv∥ ? ∥u ? v∥ for all u, v?X) and such that T(X) has a nonempty interior. Does this guarantee that T is onto? We give a counterexample in the case of X=L1(N). 相似文献
3.
For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P k (A). We show that P1(A) is a subset of P k (A) for all k. If T in P1(A), then Tn lies in P1(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P k (A), then UTV lies in P k (A) for all natural k. Let A be unital, then 1) if an element T in P1(A) is right invertible, then any right inverse element T?1 lies in P1(A); 2) for ßßIßß = 1 the class P1(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P1(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P1(A) coincides with the set of all paranormal operators on a Hilbert space H. 相似文献
4.
Given a unitary operator T in a Hilbert space H = (H, 〈·, ·〉) convergence results for two sequences of ( two-point Padé approximants to the function f(z) = 〈(I ? zT)?1u0, u0〉, (u0 ∈ H, ∥ u0∥ = 1, z regular for T) are given. An elementary proof is also given of the well-known operator version of the trigonometric moment problem, not using the solution of the classical trigonometric moment problem. 相似文献
5.
Let ∥·∥ be an operator norm and ∥·∥D its dual. Then it is shown that ∥A∥D? ∑|λi(A)|, where λi(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm. 相似文献
6.
Let {T(t)}t≥0 be a C0–semigroup on a Banach space X with generator A, and let H∞T be the space of all x ∈ X such that the local resolvent λ ↦ R(λ, A)x has a bounded holomorphic extension to the right half–plane. For the class of integrable functions ϕ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus ϕ ↦ Tϕ, as operators on H∞T. Weshow that each orbit T(·)Tϕx is bounded and uniformly continuous, and T(t)Tϕx → 0 weakly as t → ∞, and we give a new proof that ∥T(t)R(μ, A)x∥ = O(t). We also show that ∥T(t)Tϕx∥ → 0 when T is sun –reflexive, and that ∥T(t)R(μ, A)x∥ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H∞T. 相似文献
7.
J. -C. Bermond 《Discrete Mathematics》1980,30(3):295-298
Soit H = (X,F) un hypergraphe h-uniforme avec ∥X∥ = n et soit Lh±1(H) le graphe dont les sommets représentent les arêtes de H. deux sommets étant relíes si et seulement si les arétes qu'ils représentent intersectent en h ± 1 sommets. Nous montrons que si Lh±1(H) ne contient pas de cycle, alors . la borne étant exacte pour h = 2 et pour des valeurs de H pour h = 3. Ce probl`eme mène á une conjecture sur les “presque systèmes de Steine.”Let H = (X, F) be a h-uniform hypergraph, with ∥X∥ = n and let Lh±1(H) be the graph whose vertices are the edges of H, two vertices being joined if and only if the edges they represent intersect in h ±1 vertices. We prove that, if Lh±1H contains no cycle, then ; moreover the bound is exact for h = 2 and with some values of n for h = 3. This problem leads to a conjecture on “almost Steiner systems”. 相似文献
8.
9.
V.B Headley 《Journal of Mathematical Analysis and Applications》1985,108(1):283-292
Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim on an arc A of ?Δ with length . It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = supz?Δ, where C1 = limn→∞. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that almost everywhere. It is proved that inff?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C1. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥fp∥ ? p∥f∥, for any positive integer p. 相似文献
10.
Irmtraud Stephani 《Mathematische Nachrichten》1980,94(1):29-41
It is well-known that an operator T ∈ L(E, F) is strictly singular if ∥Tx∥≧λ∥x∥ on a subspace Z ? E implies dim Z < + ∞. The present paper deals with ideals of operators defined by a condition — ∥Tx∥≧λ∥x∥ on an infinite-dimensional subspace Z ? E implies Z ? F — F being a ?quasi-injective”? class of BANACH spaces. 相似文献
11.
Achiya Dax 《Numerical Linear Algebra with Applications》2004,11(7):675-692
Let x * denote the solution of a linear least‐squares problem of the form where A is a full rank m × n matrix, m > n. Let r *= b ‐ A x * denote the corresponding residual vector. In most problems one is satisfied with accurate computation of x *. Yet in some applications, such as affine scaling methods, one is also interested in accurate computation of the unit residual vector r */∥ r *∥2. The difficulties arise when ∥ r *∥2 is much smaller than ∥ b ∥2. Let x? and r? denote the computed values of x * and r *, respectively. Let εdenote the machine precision in our computations, and assume that r? is computed from the equality r? = b ‐A x? . Then, no matter how accurate x? is, the unit residual vector û = r? /∥ r? ∥2 contains an error vector whose size is likely to exceed ε∥ b ∥2/∥ r* ∥2. That is, the smaller ∥ r* ∥2 the larger the error. Thus although the computed unit residual should satisfy AT û = 0 , in practice the size of ∥AT û ∥2 is about ε∥A∥2∥ b ∥2/∥ r* ∥2. The methods discussed in this paper compute a residual vector, r? , for which ∥AT r? ∥2 is not much larger than ε∥A∥2∥ r? ∥2. Numerical experiments illustrate the difficulties in computing small residuals and the usefulness of the proposed safeguards. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
12.
A matrix seminorm ∥·∥ is called supspectral if it satisfies the condition that the spectral radius of a square matrix A is lim sup ∥An∥1/n as n→∞. This property is shown to be equivalent to each of two conditions on ∥·∥, one characterizing behavior on idempotent A, and the other characterizing behavior on non-nilpotent A. Examples of supspectral seminorms are provided. 相似文献
13.
R. Strašek 《Acta Mathematica Hungarica》2003,101(1-2):63-68
We consider the class of Euclidean algebras associated to Minkowski light cones and called Lorentz algebras. We prove that in Lorentz algebras the estimate ∥P(a,b) ∥∞≥(√2-1) ∥a∥∞ ∥b∥∞ is valid for the spectral norm and is therefore independent of the dimension of the Lorentz algebra. 相似文献
14.
15.
The Schur product of two n×n complex matrices A=(aij), B=(bij) is defined by A°B=(aijbij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on n by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥m. It is proved here that for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that and this is so iff , where ā is the matrix obtained by taking entrywise conjugates of A. 相似文献
16.
A smoothing self-adaptive Levenberg-Marquardt algorithm for solving system of nonlinear inequalities
In this paper, we consider the smoothing self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H(x) = 0. A smoothing self-adaptive Levenberg-Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg-Marquardt parameter μk is chosen as the product of μk = ∥Hk∥δ with δ ∈ (0, 2] being a positive constant. We will show that if ∥Hk∥δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈ (0, 1), while quadratically for δ ∈ [1, 2]. Numerical results show that the new method performs very well for system of inequalities. 相似文献
17.
A.M Fink 《Journal of Mathematical Analysis and Applications》1977,61(2):404-408
We show how inequalities of the type when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝0ak(s, z)F(s) ds. 相似文献
18.
《Quaestiones Mathematicae》2013,36(1-2):225-235
Abstract We say that a normed space X has the Daugavet property (DP) if for every finite rank operator K in X the equality ∥I + T∥ = 1 + ∥T∥ holds. It is known that C[0,1] and L 1[0,1] have DP. We prove that if X has DP then X has no unconditional basis. We also discuss anti-Daugavet property, hereditary DP-spaces and construct a strictly convex normed space having DP. 相似文献
19.
Frederick Bloom 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1976,27(6):853-862
For the nonlinear wave equationu tt -Nu +G(t,u, u t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ 0 T ∥G(τ,u, u τ)∥2 dτ ≤ κ ∫ 0 T ∥?(τ)∥2 dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented. 相似文献
20.
Derek W Robinson 《Journal of Functional Analysis》1977,24(3):280-290
Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space with corresponding infinitesimal generators S, T. We prove the following: ∥Ut, ? Vt ∥ = O(t), t → 0, if and only if U = V; ∥Ut ? Vt∥ = O(tα), t → 0; with 0 ? α ? 1, if and only if , where Ω, P, are bounded operators on such that if and only if has a bounded extension to 1. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras. 相似文献